# Discrete Gaussian Distribution

### Gaussian Function:
\begin{equation}
f(x) = a \cdot \exp\left(-\frac{(x - c)^2}{2\sigma^2}\right)
\end{equation}

### Gaussian Measure:

- For $a = 1$, we define the **Gaussian measure** in $\mathbb{R}$ as:
\begin{equation}
\rho_{\sigma, c}(x) = \exp\left(-\frac{(x - c)^2}{2\sigma^2}\right)
\end{equation}
- Generalized to $\mathbb{R}^n$ using $s = \sqrt{2\pi}\sigma$:
\begin{equation}
\rho_{s, c}(\mathbf{x}) = \exp\left(-\frac{\pi \|\mathbf{x} - \mathbf{c}\|^2}{s^2}\right)
\end{equation}
- The total measure over $\mathbb{R}^n$:
\begin{equation}
\int_{\mathbf{x} \in \mathbb{R}^n} \rho_{s, c}(\mathbf{x}) \, d\mathbf{x} = s^n
\end{equation}

### Gaussian PDF (Probability Density Function):
\begin{equation}
D_{s, c}(\mathbf{x}) = \frac{\rho_{s, c}(\mathbf{x})}{s^n}
\end{equation}

### Discrete Gaussian over lattice $L$:
- Gaussian measure over lattice $L$:
\begin{equation}
\rho_{s, c}(L) = \sum_{\mathbf{x} \in L} \rho_{s, c}(\mathbf{x})
\end{equation}
- Discretized density function:
\begin{equation}
D_{s, c}(L) = \frac{\rho_{s, c}(L)}{s^n}
\end{equation}
- Discrete Gaussian distribution:
\begin{equation}
D_{L, s, c}(\mathbf{x}) = \frac{D_{s, c}(\mathbf{x})}{D_{s, c}(L)}
= \frac{\rho_{s, c}(\mathbf{x})}{\rho_{s, c}(L)}
\end{equation}